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1 Strathprints Institutiona Repository Doean Maini, Victorita and Lanteri, Stephane and Perrusse, Ronan (2008) A domain decomposition method for soving the three-dimensiona time-harmonic Maxwe equations discretized by discontinuous Gaerkin methods. Journa of Computationa Physics, 227 (3). pp ISSN Strathprints is designed to aow users to access the research output of the University of Strathcyde. Copyright c and Mora Rights for the papers on this site are retained by the individua authors and/or other copyright owners. ou may not engage in further distribution of the materia for any profitmaking activities or any commercia gain. ou may freey distribute both the ur ( strathprints.strath.ac.uk/) and the content of this paper for research or study, educationa, or not-for-profit purposes without prior permission or charge. Any correspondence concerning this service shoud be sent to Strathprints administrator: maito:strathprints@strath.ac.uk

2 PAPER SESSION PA2. ID 228. A domain decomposition strategy for soving time-harmonic Maxwe s equations discretized by a discontinuous Gaerkin method Victorita Doean, Stéphane Lanteri, and Ronan Perrusse This work has been submitted to the IEEE for possibe pubication. Copyright may be transferred without notice, after which this version may no onger be accessibe. Abstract The numerica soution of the three-dimensiona time-harmonic Maxwe equations using high order methods such as discontinuous Gaerkin formuations require efficient sovers. A domain decomposition strategy is introduced for this purpose. This strategy is based on optimized Schwarz methods appied to the first order form of the Maxwe system and eads to the best possibe convergence of these agorithms. The principes are expained for a 2D mode probem and numerica simuations confirm the predicted theoretica behavior. The efficiency is further demonstrated on more reaistic 3D geometries incuding a bioeectromagnetism appication. Index Terms Domain decomposition methods, discontinuous Gaerkin methods, optimized interface conditions. I. INTRODUCTION DISCONTINUOUS Gaerkin (DG) methods are emerging for the soution of time-harmonic Maxwe s equations [] because of the enhanced fexibiity compared to the conforming edge eement method [2]. For instance, by using a DG method, deaing with non-conforming meshes is straightforward. The formuation of such methods in the case of the first order eiptic Maxwe system has been fuy anayzed in [3] and an extension to the time-harmonic first order system is introduced in [4] where a numerica comparison of different schemes is proposed. Nonetheess, before taking advantage of the fexibiity of the DG methods, the design of efficient agorithms for the resuting sparse inear system has to be addressed. Here we propose a domain decomposition (DD) strategy based on optimized Schwarz methods [5], [7], [9]. First, the DD strategy is introduced in the two-domain case for a 2D transverse eectric mode probem. Then the discretization of the probem by a DG method is presented. Finay, numerica resuts for a 2D probem confirm the expected theoretica behavior of the DD method and 3D numerica experiments on simpified probems pave the way for more reaistic appications. V. Doean is with J. A. Dieudonné Mathematics Laboratory, UMR CNRS 662, University of Nice-Sophia Antipois, F-0608 Nice cedex, France (emai: doean@unice.fr). S. Lanteri is with INRIA, 2004 Route des Lucioes, BP 93, F Sophia Antipois cedex, France (e-mai: stephane.anteri@inria.fr). R. Perrusse is with the Laboratoire Ampère, UMR CNRS 5005, and Université de Lyon, Écoe Centrae de Lyon, F-6934 Écuy cedex, France (e-mai: ronan.perrusse@ec-yon.fr). Manuscript received June 24, 2007; revised XX. II. THE DOMAIN DECOMPOSITION STRATEG For the sake of simpicity we consider the foowing nondimensioned transverse eectric mode probem in a domain Ω R 2 : Find the eectromagnetic fied (E, H) satisfying: iωεe cur H = 0, in Ω, iωµh + cure = 0, in Ω, n (E E inc ) + (H H inc ) = 0, on Ω. The parameters ε and µ denote respectivey the reative dieectric permittivity and the reative magnetic permeabiity, ω the anguar frequency, n the unitary outgoing norma and (E inc, H inc ) the components of an incident eectromagnetic wave. For soving (), the domain Ω is decomposed in two nonoverapping subdomains Ω and Ω 2. The common interface to Ω and Ω 2 is denoted by Γ. The DD strategy is then a variant of the cassica Schwarz method: We start with an initia eectromagnetic fied (E 0, H0 ) on each subdomain Ω, =, 2. The (p + )-th iterate (E p+, H p+ ) is the soution of () restricted to the subdomain Ω with an interface transmission condition on Γ of the form: { n (E p+ E p m ) + S (H p+ H p m ) = 0, with S = α + β τ, 2 (2) where 2 τ denotes the second-order derivative aong the interface. The operator S ensures the transmission of the fied (E p m, H p m) computed at the previous iteration in the neighboring subdomain m with parameters α, β propery chosen to contro the convergence of the agorithm. The imit of the sequence (E p, Hp ) p N is the restriction to Ω of (E, H) the soution of (). Thus, we can use a stopping criterion: 2 = (E p+ (), H p+ ) (E p, Hp ) (E, H ) (E0, H0 ) < to, (3) where to is the prescribed accuracy and a norm. Després in [6] was the first to use to use this strategy for time-harmonic equations with the choice S =, for =, 2 which coincides with a first order absorbing boundary condition. However, the convergence rate of the iterative process with this boundary condition is strongy dependent on

3 PAPER SESSION PA2. ID the mesh size used for the discretization and the convergence to the soution can be sow. We refer to this condition as the cassica condition in the foowing. Nonetheess, it is possibe to greaty improve the convergence rate by optimizing it with respect to α i, β i. This theoretica study is done in [7] directy on () and in [8] for the second order cur cur formuation. The cosed-form expressions obtained for the coefficients α i, β i are in particuar dependent of the mesh size. These expressions are then used in a DD strategy generaized to more than two subdomains. Let us briefy reca the anaysis proposed in [7] in order to deduce a theoretica convergence rate ρ for the iterative agorithm depending on the mesh size h. This anaysis is done on the continuous, i.e. without discretization, DD method but numerica arguments are given in the foowing for showing that the resuts remain vaid with a discretization. The study of ρ is done for a decomposition of R d into two infinite domains. A Fourier transform is appied with respect to the tangentia variabes to the interface (artificia boundary separating the two domains). The resuting oca equations can be soved eading to the formuation of an iterative process appied to the interface variabes. Then, we obtain the reduction factor of the error (ρ) as a function of the Fourier variabe and the parameters invoved in the interface conditions. In order to obtain the best possibe convergence rate, one needs to optimize this quantity with respect to the parameters, for the range of possibe spatia frequencies that can be represented on a given mesh. In the seque, we treat the cases of zero order boundary conditions where we take in (2) β equa to zero that is the case of generaized impedance conditions. Two possibiities are considered: α = α 2 and α α 2. It has been proved that α are equa to (iω) (p + ip ) where p and p 2 are reported in Tabe I. When the mesh parameter h is sma, the maximum numerica frequency that can be represented on the mesh is estimated by k max = C h where C is a constant. We aso define k ± such that k ω k + in order to excude the frequency ω from the optimization process and this frequency being treated by the Kryov method (see aso [9] for detais). TABLE I CONVERGENCE RATE AND TRANSMISSION CONDITION PARAMETERS. Case ρ p p 2 2C 4ω π h πc 4ω 2 h 2 C 8ω π h 4 4 π 4 C 8 ω 2h 4 πc 4ω 2 h π 4 3 C 8 ω h 3 4 III. DISCRETIATION OF THE PROBLEM For the discretization of the oca probem on Ω i, a DG method is used. The domain is decomposed into a set of eements T h such that K Th K = Ω i. The approximate soution (E h, H h ) of () is an eement of V 3 h where V h is the finite eement space of square-integrabe discontinuous scaar fieds whose restriction to an eement K is poynomia of degree k: V h = { V L 2 (Ω) / K T h, V K P k (K) }. (4) Thus no particuar continuity constraint is enforced at the interface of each eement. The weak formuation of the discrete probem is then the foowing: Find (E h, H h ) Vh 2 V h such that: ( a(h h, G) + b(g,e h ) = H inc NnE t inc) Gds, b(h h,f) c(e h,f) = F Vh 2,G V h. Ω Ω 2 2 (N nh inc N n NnE t inc ) t Fds, where: a(h h, G) = iωµh h Gdv + αh h Gds Ω h F Γ 0 F + Ω 2 H hgds, b(g,e h ) = G cur(e h )dv {G} t E h T ds K T h Ω 2 G(Nt n E h)ds. and finay: c(e h,f) = iωǫe t h Fdv + Ω h F Γ 0 + Ω 2 (Nt ne h )(NnF)ds. t K F F Γ 0 F α E h t T F Tds The matrix N t n denotes ( n y n x ), for a vector n of R 2, {G} and F T denotes respectivey the mean of G and the jump of the tangentia component of F over an interior face F. Note that in order to keep the consistency with the continuous probem, a numerica fux is defined on the interface of each eement enforcing weaky the tangentia continuity constraint for the eectric and magnetic fied. The proper choice of different kind of fuxes has been discussed from the numerica point of view in [4]. For instance, the choice α = 0 corresponds to the case of the centered fux, and α = /2 to the choice of a simpified upwind fux. The former has the advantage to be easy to impement and requires a ower memory storage. The atter has better convergence properties. IV. TWO-DIMENSIONAL NUMERICAL RESULTS The agreement between the theoretica and numerica convergence rates is demonstrated on a probem with Ω =]0; [ 2, (E inc, H inc ) = exp( iωx)(0,, ) and ω = 2π discretized by discontinuous Gaerkin methods. Firsty, we study the infuence of the choice of the numerica fux and of the poynomia order on the theoretica convergence rate. Secondy, we study the mutidomain case. A. Infuence of the eement order and of the numerica fux The first DG discretization is based on a trianguar uniform mesh with P (K) as the oca space in each eement K. On Figure, the number of iterations for achieving a prescribed accuracy against the mesh size is shown for both boundary conditions (Case and Case 2). The curves fit nicey the

4 PAPER SESSION PA2. ID iterations Number of iterations against the mesh size. ω = 2 π h 0.5 Parameters Case h 0.25 Parameters Case 2 we make use of GMRES for soving the interface system. We observe that, independenty of the fux chosen, the number of iterations grows roughy as Ns 0.3 where N s is the number of subdomains. Thus, the convergence deteriorates with the number of subdomains and it advocates for the use of a coarse grid in order to obtain an optima sover. Nevertheess the hierarchy of the transmission conditions is maintained and the optimized versions enabe us to acceerate the convergence compared to the cassica condition [6]. Fig h Number of iterations against the mesh size h. Logarithmic scae. dependence in h predicted by the theory i.e. they behave ike h 0.5 for Case and ike h 0.25 for Case 2. In order to demonstrate that the theoretica resuts are independent of the choice of the DG discretization, we aso approximate numericay this asymptotic convergence rate using three poynomia orders for the eement interpoation (here quadriatera eements are considered to simpify the management of the muti-domain case for this academic probem) and two different fuxes; these experiments are reported in Tabe II. A behavior cose to h 0.5 and h 0.25 for the number of iterations is obtained by the numerica experiments, independenty of the choice of the numerica fux and of the numerica order. TABLE II ESTIMATED VALUE OF δ WHERE ρ = Ch δ. Fux Q 0 Q Q 2 Centered Case Centered Case Upwind Case Upwind Case Number of iterations 0 2 Cassica Est. exponent: 0.29 Case Est. exponent: 0.26 Case 2 Est. exponent: Number of subdomains Fig. 2. Number of iterations against the number of subdomains. The number of dof is constant. Resuts for the centered fux. Number of iterations 0 2 Cassica Est. exponent: 0.3 Case Est. exponent: 0.28 Case 2 Est. exponent: 0.3 B. Infuence of the number of subdomains As in practice more than two domains are used, the performances of the optimized conditions are evauated for more subdomains. The same probem as in Subsection IV-A is soved. The numerica experiments are sti performed on a structured grid and then the partition into severa subdomains is made in a reguar way: a decomposition in N N rectanguar sudomains of the unit square. The resuts for the centered case are shown on Figure 2 and the resuts for the upwind case on Figure 3; a Q poynomia approximation is used for these figures. Note that for obtaining these resuts, a Kryov subspace method is couped to the Schwarz agorithm. Indeed, as it is expained in [5], the DD method can be formuated as a inear system whose unknowns are auxiiary interface variabes. This interface system is usuay soved by a Kryov method which gives more robustness to the DD strategy. Here Number of subdomains Fig. 3. Number of iterations against the number of subdomains. The number of dof is constant. Resuts for the upwind fux. V. THREE-DIMENSIONAL PROBLEM A. Scattering by a sphere The impementation of optimized interface conditions for three-dimensiona time-harmonic Maxwe s equations is a work in progress. Here, we give preiminary resuts for the DD strategy based on first order absorbing boundary conditions as transmission conditions. The probem under consideration is the scattering of a pane wave by a perfecty conducting unit sphere. The incident wave is given by E inc = (exp( iωx), 0, 0) t and H inc =

5 X X PAPER SESSION PA2. ID (0, exp( iωx), 0) t, with ω = 4π. The absorbing boundary is set to one waveength from the surface of the perfecty conducting sphere. The mesh is composed of,382,400 tetrahedra and a P 0 (K) oca space is used for the DG method. The tota number of unknowns is 8,294,400. Numerica experiments are conducted on a custer of 64 AMD Opteron/2 GHz processors with a Gigabit Ethernet interconnection. One subdomain is associated to each processor and a sparse matrix direct method is used to sove the subdomain probem. Note that we use a BiCGstab() method [0] either for soving the interface system or as a goba sover without preconditionner. Performance resuts are given in Tabe III where DDM refers to the DD soution strategy. The per processor time for performing the factorization is 8.0 sec (min)/02.0 sec (max) whie the associated memory usage is 405 MB (min)/00 MB (max). In addition to the gain in computing time, a cear advantage of the DD strategy is its parae efficiency that can be evauated here as the ratio of CPU (max) over Eapsed which is equa to 92% whie the corresponding feature for the goba sover is 74%. Performance resuts are given in Tabe V. The parae efficiency, evauated using the maximum CPU to REAL ratio, ranges from 65% to 75%. TABLE V COMPUTATION TIMES. U: UPWIND FLUX, C: CENTERED FLUX. Mesh Method N s # it CPU (min/max) REAL M DG-P -c sec/466 sec 74 sec - DG-P -u sec/547 sec 765 sec M2 DG-P 0 -c sec/322 sec 428 sec X RE X TABLE III PERFORMANCE RESULTS. CPU (MIN/MAX) ARE PER PROCESSOR MEASURES OF THE CPU TIME. ELAPSED IS THE ELAPSED TIME. Fig. 4. View of the mesh M and a soution computed on this mesh. Sover # iter CPU (min) CPU (max) Eapsed Goba sec sec sec DDM sec 43.0 sec sec B. A bioeectromagnetism exampe We concude this section of resuts with the appication of the proposed numerica methodoogy to the simuation of a time-harmonic eectromagnetic wave propagation probem in an irreguary shaped and heterogeneous medium. The probem under consideration is concerned with the propagation of a pane wave in reaistic geometrica modes of head tissues. Two tetrahedra meshes have been used whose characteristics are summarized in Tabe IV. The frequency of the incident TABLE IV CHARACTERISTICS OF THE TETRAHEDRAL MESHES. Mesh # tetraheda L min (m) Lmax (m) Lavg (m) M 36, M2,853, pane wave is F=800 MHz and its poarization is such that: k = (k x, 0, 0) t, E = (0, 0,E z ) t and H = (0,H y, 0) t. The eectromagnetic parameters of the materias are set to artificia vaues for the purpose of exempifying the characteristics of the propagation of the pane wave in the head tissues (nu conductivity, ε r = 4.0 for the brain, ε r = 6.5 for the cerebrospina fuid, ε r =.5 for the sku and ε r = 4.0 for the skin). For the computations reported here, the DG methods with P 0 and P eements and upwind and centered fuxes are used for the meshes M and M2. The mesh M and a view of the soution is proposed on Figure 4. VI. CONCLUSION In this paper, cassica and optimized Schwarz agorithms have been appied to time-harmonic Maxwe s equations discretized by DG methods. Concerning Schwarz agorithms based on optimized interface conditions, two-dimensiona numerica resuts show a good agreement with the theory in the case of the simpest optimized conditions and the behavior is independent of the choice of the fux and the poynomia order for the finite eement space. Preiminary resuts in the three-dimensiona case are very promising for cassica interface conditions, opening the way to improvements by using optimized interface conditions on reaistic appications. REFERENCES [] P. Houston, I. Perugia, A. Schneebei, and D. Schötzau, Interior penaty method for the indefinite time-harmonic Maxwe equations, Numer. Math., vo. 00, no. 3, pp , [2] P. Monk, Finite eement methods for Maxwe s equations, ser. Numerica Mathematics and Scientific Computation. New ork: Oxford University Press, [3] A. Ern and J.-L. Guermond, Discontinuous Gaerkin methods for Friedrichs systems I. Genera theory, SIAM J. Numer. Ana., vo. 44, no. 2, pp , [4] V. Doean, H. Fo, S. Lanteri, and R. Perrusse, Soution of the timeharmonic Maxwe equations using discontinuous Gaerkin methods, Accepted for pubication in Journa of Computationa and Appied Mathematics., 2007, [5] M. J. Gander, Optimized Schwarz methods, SIAM J. Numer. Ana., vo. 44, no. 2, pp (eectronic), [6] B. Després, Méthodes de décomposition de domaines pour es probèmes de propagation d ondes en régime harmonique. Le théorème de Borg et équation de Hi vectoriee, Ph.D. dissertation, Université Paris 9, France, 99. [7] V. Doean, M. J. Gander, and L. Gerardo-Giorda, Optimized Schwarz methods for Maxwe s equations, submitted. Preprint avaiabe at http: //ha.archives-ouvertes.fr/ha [8] A. Aonso Rodríguez and L. Gerardo-Giorda, New nonoverapping domain decomposition methods for the harmonic Maxwe system, SIAM J. Sci. Comput., vo. 28, no., pp (eectronic), 2006.

6 PAPER SESSION PA2. ID [9] M.-J. Gander, F. Magouès, and F. Nataf, Optimized Schwarz methods without overap for the Hemhotz equation, SIAM J. Sci. Comput., vo. 24-, pp , [0] G. L. G. Seijpen and D. R. Fokkema, BiCGstab() for inear equations invoving unsymmetric matrices with compex spectrum, Eectron. Trans. Numer. Ana., vo., no. Sept., pp. 32 (eectronic ony), 993.